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User blog:All Things Logical/What is mathematics really built upon?
What is mathematics really built upon? It is not numbers, not sets either, it's vectors. Using vectors to create numbers We can define 1 to be a unit vector in the x direction. We get for free -1, which is a unit vector in the opposite direction. We can now define all the real numbers. Operations are easy. Addition is translating the vectors. Let us say you wanted to know what 3 + 5 was. You would shift 0 to 3, and then count 5 units forward. If you add by -1, you would translate the vectors backwards by -1. Multiplication is stretching vectors. Let us say you wanted what to know what 3 * 5 was. You would stretch the one vector until it's 3 times as long, and see where 5 landed. If you multiply by -1, you would flip the number line. Exponentiation is on a logarithmic scale. Let us say wanted to know what 3^5 was. You would have a logarithmic scale, and start from one, and jump 3 units logarithmically 5 times. When you do this, you get to 243, which is 3^5. etc... You see that the length of the vectors is dependent of the previous operation. For multiplication, vectors were increasing by +3. For exponentiation, the vectors were increasing by x3, etc... Since i is sqrt(-1), then it is a pi/2 rotation. Using vectors to create shapes A shape can be described as the vectors that define its border. For example, a square could be described with 4 vectors, a triangle 3, and so on. Shape subtraction is quite easy to define, since it is the shape that a certain group of vectors define, without another shape that a different group of vectors define. Shape addition is quite easy also, since it is the shape that a certain group of vectors define, with another shape a certain group of vectors define. There may be overlap, but that's okay. The shapes could also be light years apart. Shapes like the circle are difficult to define, since you need an infinite amount of vectors. All that matters is you can define vectors, given that the vectors are infinitely small. Shape translation can be though of as a shape, which we know can be written as vectors, and a vector to tell the shape to move somewhere. Shape rotation can be described as a shape which can be written as vectors, and circle, which can also be written as vectors. Translation, rotation, and rotation + translation are the only three ways a shape can turn into a congruent shape. They can all be written in terms of vectors. Shape scaling can be though of as a shape, and a number. Both of which can be represented using vectors. Vectors that don't close up still make a shape, just a really thin one. Using vectors to create sets We know how to create numbers, and shapes using vectors. A set is just a collection of these things. You can even have a set of vectors, if you want. We have been indirectly creating sets out of vectors, now the final part is to put it all together. Since each element can be described using vectors, the whole thing can be described using vectors. Category:Blog posts